Some places have specific programs , e.g.
http://www.azhumane.org/our-services/animal-rescue-eamt/
Although the training is usually not standardized (ie certificate and all that) and lot of the caseload are cruelty/abuse/stray cases. They tend to work closely with animal control and local hospitals.
People training to be vet assistants can usually find some basic first aid courses and things like that. However, I think that
most emergency work training is generally directed towards students, vet technicians, and vets themselves.